Vladimir L. Popov: Generically multiple transitive algebraic group actions

popovvl@orc.ru

Submission: 2004, Sep 1, revised 2005, Mar 5

With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$, called the generic transitivity degree of $G$, equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open $G$-orbit. We calculate ${\rm gtd}(G)$ for all reductive and all solvable linear algebraic groups $G$. In particular, we show that $G$ is abelian if and only if ${\rm gtd}(G)=1$. We prove that if $G$ is a nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$, where $P$ is a proper maximal parabolic subgroup of $G$ (but in general not only for such $X$). For every reductive group $G$ and every proper maximal parabolic subgroup $P$ of $G$, we find the maximal positive integer $d$ such that the diagonal action of $G$ on $(G/P)^d$ admits an open $G$-orbit. In particular, we classify all proper maximal parabolic subgroups $P$ of $G$ such that the action of $G$ on $(G/P)^3$ admits an open $G$-orbit, answering a question of {\sc M.~Burger}

2000 Mathematics Subject Classification: 14L30, 14L35, 14L40, 20G05

Keywords and Phrases: Algebraic group, action, reductive group, orbit, stabilizer, highest weight

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